Lens design method and resulting aspheric lens

ABSTRACT

An aspheric lens for providing improved vision and a method for generating such a lens is described. The lens provides a sharp image focus while minimizing image aberrations. The method utilizes ray tracing techniques in conjunction with Modulation Transfer functions to accurately account for the total corrective lens-eye system. The lens may be in the form of a contact lens, an intraocular lens, a natural lens or a spectacle lens, and is suitable for correcting myopia, presbyopia, astigmatism and other focusing problems. The lens is characterized by a hyperbolic or parabolic surface which functions to reduce spherical aberrations and minimize the retinal image spot size.

This application is a continuation-in-part of application of Ser. No.557,261 filed Jul. 24, 1990, now U.S. Pat. No. 5,050,981.

BACKGROUND OF THE INVENTION

This invention is a method for designing a lens to provide an optimalcorrective lens-eye system having minimal image aberrations and theresulting lens having an aspheric surface for use as an contact,intraocular or spectacle lens, particularly a lens in which the surfacehas a hyperbolic or parabolic curvature.

The curvature of a conventional lens surface may be described in termsof "conic sections." The family of conic sections includes the sphere,parabola, ellipse, and hyperbola. All rotationally symmetric conicsections may be expressed in terms of a single equation: ##EQU1## whereX is the aspheric surface point at position Y, r is the central radius,and the kappa factor, κ, is the aspheric coefficient.

Other conic constants or aspheric coefficients include the eccentricity,e, which related to κ by the equation κ=-e², and the rho factor, ρ,defined as (1-e²).

The value of the aspheric coefficient determines the form of the conicsection. For a sphere, e=0 and κ=0. An ellipse has an eccentricitybetween 0 and 1 and a κ between 0 and -1. A parabola is characterized byan e=1 (κ=-1). For a hyperbola, e is greater than 1 and κ is less thannegative one.

Conventionally, most lens surfaces are spherical or near-spherical incurvature. Theoretically, for an infinitely thin lens, a sphericalcurvature is ideal to sharply focus the light passing through the lens.However, the curvatures and thicknesses of a real lens producewell-known optical aberrations, including spherical aberration, coma,distortion, and astigmatism; i.e., light from a point source passingthrough different areas of the lens that does not focus at a singlepoint. This causes a certain amount of blurring. Furthermore, purelyspherical lenses are not suitable for correcting astigmatic vision orfor overcoming presbyopia.

For this reason, many different types of lenses have been designed forthe purpose of minimizing spherical aberration, correcting ocularastigmatism, or providing a bifocal effect that allows thenonaccommodative eye to see objects both near and far. Unfortunately,current designs suffer from serious drawbacks, such as producing blurredor hazy images, or inability to provide sharp focusing at every visualdistance.

Aspheric lenses having elliptical surfaces have been used to reduceoptical aberrations. Some well known examples are the use of parabolicobjective mirrors in astronomical telescopes and the use of ellipses oflow eccentricity to correct for aberrations of a contact lens.

The design of an aspheric lens in isolation is well known. There are avariety of commercially available software packages that use variationsof the above equation to generate aspheric lens designs. An example ofthese are: Super OSLO by Sinclair Optics, Inc., Code-V by OpticalResearch Associates and GENII-PC by Genesee Optics, Inc. These opticaldesign programs are the most widely used packages available. Despite thedifferent approaches used by the three methods, all packages haveyielded identical results in aspheric lens design calculations. Whenused alone for vision correction, carefully designed elliptical lensesdo provide an improved focus. However, when used in a system includingthe human eye, elliptical lenses are not significantly better thanspherical lenses. This is because the eye contains a greater amount ofaberration than the elliptical lens is able to correct as part of theoverall corrective lens-eye system.

Methods used in the past to produce corrective lenses for the eye haveresulted in lenses that are non-spherical. In U.S. Pat. No. 4,170,193 toVolk a lens is described which corrects for accommodative insufficiencyby increasing dioptric power peripheralward. While this lens and otherprior lens designs are not strictly spherical, it is not a pure asphere,and includes higher order deformation coefficients. This yields asurface which is radically different than that proposed herein. Aflattening curve, such a hyperbola, would show a slight dioptricdecrease peripheralward. Prior lens designs, while attempting to solvevarious optical problems by varying from a strictly spherical lensdesign, do not strive for improved vision by reducing the aberration ofthe image that strikes the retina of the eye.

An important reason for the common use of lens designs that have theabove-noted limitations is the failure to take into account the effectsof the entire lens eye system. Lenses are usually designed as if thelens would be the only element that contributes to image aberrations,but there are may elements in the eye that affect image focus, such asthe surfaces of the cornea and of the eye's natural lens. While theelliptical form was useful in reducing aberrations of the lens itself,when the lens is placed into a system containing all of the refractingsurfaces of the human eye additional aspherical correction is required.

SUMMARY OF THE INVENTION

The present invention is that this required correction has been found tobe in the form of certain ellipse or a parabola and provides a lens foreffectively focusing light on the retina of the eye and a method forproducing such lens. The lens has a rotationally symmetric asphericsurface in the form of a ellipse or parabola defined by the equation:##EQU2## where X is the aspheric surface point at position Y, r is thecentral radius, and κ is a commonly used aspheric constant, wherein thevalue of κ is less than or equal to -0.5.

It is an object of the present invention to provide a method for thesystematic approach to the design of an aspheric lens in which the lensis considered and optimized as part of the entire corrective lens-eyesystem.

It is a further object of the present invention to use the modulationtransfer function (the modulation scale from black and white to gray)and the spatial frequency (showing the degree to which objects ofincreasing spatial frequency can be resolved) to optimize a correctivelens design when considered with the corrective lens-eye system.

An additional object of the present invention is to provide a methodthat produces a lens that optimizes the focusing of an image on theretina of the eye and that minimizes image aberrations and blurring.

It is an object of the present invention to provide a novel asphericlens design suitable for use in a contact lens, an intraocular lens, ora spectacle lens.

It is also an object of the present invention to provide a lens for useon the surface of, in or near the human eye wherein a lens surface iscurved in the shape of a hyperbola.

It is a further object of the present invention to provide a lens foruse on the surface of, in or near the human eye wherein a lens surfaceis curved in the shape of an ellipse or a parabola.

Another object of this invention is to provide an aspheric lens suitablefor use by those suffering from presbyopia, myopia, hyperopia,astigmatism, or other vision focusing deficiencies.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a front elevation of a contact lens according to the presentinvention.

FIG. 2 is a cross sectional view of the lens shown in FIG. 1 taken alongthe line 2--2.

FIG. 3 is a front elevation of an intraocular lens according to thepresent invention.

FIG. 4 is a cross section view of the lens shown in FIG. 3 taken alongthe line 4--4.

FIG. 5 graphically compares the size of the retinal image of a pointlight source as a function of pupil diameter for a myopic eye/hyperboliccontact lens system to that of a myopic eye/spherical contact lenssystem and an emmetropic eye, where each lens has the optimum opticalpower to correct the myopia of the eye.

FIG. 6 shows the best focus position relative to the retina for theimages of FIG. 5.

FIG. 7 graphically compares the curvature of a spherical surface and anaspheric surface having the same central or apical radius.

FIG. 8 is a typical Modulation Transfer Function graph showing theresolving power of the eye with a conventional corrective lens and theinherent limit of resolving power due to diffraction limits.

FIGS. 9A through F compare the modulation transfer frequency to thediffraction limit in a lens-myope system. Each figure presents thecomparison for a particular kappa factor, ranging from κ=0 in FIG. 9A toκ=-2.5 in FIG. 9F.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention applies optical ray trace techniques to an opticalschematic of the human eye to achieve heretofore unobtained performancefrom a corrective len-eye system. The human eye model was developedafter an extensive literature search on the subject of human ocularphysiology, physiological optics and anatomy. In particular, a startingpoint for the model were the Gullstrand (1862-1930) Schematic Eyes.Gullstrand created these models on the basis of available data on theanatomy of the eye generated by himself as well as other researchers.The Gullstrand eyes contain centered, spherical surfaces, and were usedthroughout the 20th century to evaluate first order (i.e., location, notlevel of aberration) image formation of the human eye.

It is recognized that there are individual variations from the averageswhich Gullstrand presented, and in addition, advances in metrologyallowed analysis in greater detail of the refractive index distribution,as well as variations in aspheric curvature of the various elements.Using the Gullstrand Schematic as a starting point, with the addition ofmore modern knowledge about the anatomy of the eye, a composite eyemodel was generated.

To first order, the model can be looked at as a three lens compoundsystem, the lenses being the corrective lens devices, the cornea, andthe crystalline lens of the eye. This can be further broken down tocontain 13 surfaces for the purpose of ray trace analysis. Thesesurfaces are:

1] Object

2] Front surface of the corrective lens

3] Back surface of the corrective lens

4] Tear layer

5] Corneal epithelium

6] Corneal endothelium aqueous interface

7] Pupil in aqueous

8] Lens anterior cortex

9] Lens anterior core

10] Lens posterior core

11] Lens posterior cortex

12] Vitreous

13] Retina

It is not usual that the image falls on the retina. Indeed, this is thedefinition of refractive error. Using ray trace techniques, the actualposition relative to the retina and quality of the image can bedetermined.

FIGS. 1 and 2 illustrate one embodiment of a lens 1 according to thisinvention which is suitable for use as a contact lens. This lens 1 has arotationally symmetric hyperbolic surface 2 and a concave sphericalsurface 3. The spherical surface 3 has a radius of curvature whichconforms to that of the outer surface of the human eye so that the lens1 may rest comfortably on the eye surface. The size of the contact lens1 should be suitable for the intended use, e.g., about 12-15 mm indiameter and no more than about 0.050-0.400 mm thick.

FIGS. 3 and 4 illustrate an intraocular lens 4 according to thisinvention. This lens 4 has a rotationally symmetric hyperbolic surface 5and a convex spherical surface 6. The intraocular lens 4 should beapproximately 4-7 mm in diameter and have a maximum thickness of about0.7-1.0 mm.

The lenses of this invention are not limited to the physical dimensionsgiven above; these dimensions are only rough guidelines. A lens may bewhatever size is suitable for the intended use.

A lens according to this invention may have two symmetric asphericsurfaces rather than one, but at least one surface must be a symmetricasphere as defined by the following equation: ##EQU3## where X is theaspheric surface point at position Y, r is the central radius and thekappa factor, κ, is a commonly used aspheric constant, wherein the valueof κ is less than or equal to 0.5. Preferably, the curvature ishyperbolic, i.e., κ is less than negative one, although a paraboliccurvature (κ=-1) or an elliptical curvature (-0.5>κ>-1) is also withinthe scope of the invention. The aspheric surface may be convex orconcave; where there are two aspheric surfaces, each may independentlybe convex or concave.

The lens of the present invention minimizes the optical aberrations ofthe lens/eye system. This produces a sharper focus on the retina, asillustrated in FIG. 5. FIG. 5 was generated by computer ray tracingmethods, and shows that the blur spot size at the retina is much smallerfor a myopic eye corrected with a hyperbolic front curve than for eitheran emmetropic (i.e., normal) eye or a myopic eye corrected by aspherical lens.

Furthermore, the light tends to be more accurately focused on theretina, as shown in FIG. 6. FIG. 6 was generated by a computer ray tracesimultaneously with FIG. 5 and shows the position of the focused imageis closest to the retina for the hyperbolic lens/eye system.

As a direct result of these advantages, a lens according to the presentinvention can provide acceptable vision for those who suffer fromastigmatism or presbyopia. The usual approach to correcting astigmatismis to provide a corrective lens that is radially asymmetric incomplimentary compensation for the radial asymmetry in either thenatural eye lens or in the retina. This approach requires the productionand inventory of a large number of lenses to suit not only the basicprescription, but also to provide the complimentary radial asymmetric ofthe eye. Further, the lens must have a means for maintaining its radialposition with respect to the eye in order that the radial variation ofthe lens matches the eye's radial requirements. Means developedheretofore have not performed with total satisfaction.

Compensation for the non-accommodating natural eye lens is traditionallyprovided by having a divided lens, with two or more focal lengths toprovide far and near vision or, as in some recent designs, a diffractiveor refractive lens with two or more focal lengths that can provideadequate near and far vision. This type of system, however, divides theincoming light among the various foci and presents each focus at everypoint on the retina. Obviously this results in a reduction in the amountof light available for any individual focus and in competing images ateach point on the retina.

The aspheric lens does not provide visual compensation to the astigmator presbyop by graded power or multiple focal lengths, but improves thecorrective lens/eye system to the point where, despite the variationscaused by asigmatism or presbyopia, the overall performance falls withinor near the range of visual acuity of the normal individual.

This occurs because the aforementioned spot size of each point fallingon the retina is reduced below that possible by the unaided emmetropiceye alone which contains a natural spherical lens. Because of theoptical superiority of the aspheric corrective lens/eye system, the blurof a point on the retina introduced by presbyopia or astigmatism isoffset by the aspheric improvement and is thereby less than (or in therange of) that found in the normal eye.

With the proper prescription, virtually any focusing deficiency may becorrected by this lens. Typically, a lens according to the presentinvention will have an optical power between about +20.00 and about-20.00 diopters.

FIG. 7 illustrates the difference between an aspheric curve 10 asdefined in the above equation and a spherical curve 11, where bothcurves have the same apical radius, r. For a given distance from apex12, x_(a) or X_(s), there is a point Y_(a) on the aspheric curve 10 anda point y_(s) on the spherical curve 11. The further X_(a) or X_(s) isfrom the apex 12, the greater the difference y_(s) -Y_(a).

A lens having the aforesaid properties is designed by a method whereinray tracing techniques are used to calculate the path of light raysthrough a corrective lens/eye system, using a sophisticated mathematicalmodel of a human eye and a corrective lens. The thickness, curvature,and material-dependent refractive index of the lens is variedmathematically and ray tracing calculations are performed on eachvariation to find the optimal lens for a given eye. The optimal lens isone which results in a sharp focus and a minimum of image aberrations.It has been found that in most cases the optical lens will have a kappafactor in the range of about -0.5 to about -2.

Image analysis involves the tracing of a large number of rays through anoptical system. The fundamental equation for tracing a ray, i.e.,determining the angle of the ray and its position) from one opticalmedium to another, via an interface between the media, is by the classicand fundamental Snell's Law equation: n₁ sin θ₁ =n₂ sin θ₂. For a systemof 13 surfaces, this can be very time consuming for even a single ray.Multiple ray analysis using several hundred rays takes a considerablenumber of operations for even a simple single element lens.

Images can be analyzed in a number of different ways. The classicalSeidel aberrations, or reductions in image quality can be calculated bytracing only a few rays. A widely accepted method of quantifying imagequality is the MTF, or Modulation Transfer Function. This can be thoughof as an extension of previous limiting resolution methods.

Referring to FIG. 8, MTF provides modulation, or contrast, resolution(measured from zero to one) versus spatial frequency or fine detail sizeof an object. The typical Modulation Transfer Function graph shown inFIG. 8 depicts the resolving power of an optical system consisting of aseries of lenses, e.g., the human eye with a corrective lens, with thattheoretically achievable.

The object bars below the X-axis show, from zero to the cutofffrequency, bars with increasing spatial frequency. The zero to one scaleon the Y-axis is the measure of resolution of the bars by an opticalsystem and that theoretically achievable at the diffraction limit. At aY value of one, the bars are sharply distinguished into black and whiteimages. As the Y value decreases, there is increasing "graying" of whiteinto black of the images. Ultimately at a Y value of zero the barscannot be distinguished at all.

The modulation can be determined by calculating the graying of the blackand white bars at each spatial frequency into a maximum and minimumlevel. The MTF modulation is the (max-min)/(max-min) contrast. The MTFwill be limited in value to a certain level called the "diffractionlimit", which would be that level of modulation contrast achievable by aperfect optical system.

The resolving power of an optical instrument of any type is defined as ameasure of the sharpness with which small images very close together canbe distinguished and is directly proportional to the diameter of theobjective aperture and inversely proportional to the wavelength of thelight. The interference pattern resulting from rays passing throughdifferent parts of an opening or coming from different points around anopaque object and then unite at a point is the manifestation ofdiffraction. Diffraction and interference effects are characteristic ofall wave phenomena. Diffraction thus limits the resolving power of alloptical instruments.

When bars of black and white are coarse and widely spaced, a lens has nodifficulty in accurately reproducing them. But as the bars get closertogether diffraction and aberrations in the lens cause some light tostray from the bright bars into the dark spaces between them, with theresult that the light bars get dimmer and the dark spaces get brighteruntil eventually there is nothing to distinguish light from darkness andresolution is lost.

MTF is calculated by tracing a large number of rays through the system,and evaluating the distribution density of these rays in the imageposition. The rays at this image position are located in the image"spot". The smaller the spot size, the better the image. The method bywhich the spot diagram is transformed to the MTF is as follows: theimage of a point object is called a point spread function, since someblurring has occured in passing through the system. The image has thusspread. By applying a Fourier Transform function to the point or spotspread function, a graph of the MTF is generated. The MTF frequency goesfrom zero ("DC" in electrical engineering terms) to the maximum orcutoff frequency, beyond which the object cannot be resolved in theimage.

Optical systems can be optimized by varying the thickness, curvature,surface asphericity, material etc. of one or several surfaces. Knownnumerical methods using computers allow rapid evaluation of the resultof varying these parameters, in terms or aberration, spot size or MTF.

This design method requires an analysis of the density of the rays inthe image position. This analysis is done by using a Fourier Transformfunction to generate modulation transfer frequencies. A computer is usedto allow the necessarily vast number of calculations to be performed ina reasonable time period. An example of the results of such calculationsis presented in FIGS. 9A through 9F. These Figures compare themodulation transfer frequency to the diffraction limit in a myopiceye-lens system, with each figure showing he results for a differentlens curvature. These results indicate that the best lenses are thosehaving a surface where κ is between -0.5 and -2.

For the human eye/corrective lens model, one is constrained to changesin the corrective lens.

When used as a contact lens, the present invention preferably comprisesa convex aspheric front surface and a concave spherical back surfacethat conforms to the curvature of the eye for a comfortable fit.

When in the form of an intraocular lens, the lens preferably will haveone convex aspheric surface. The opposite surface preferably will beplanar, concave spherical, convex aspheric, concave aspherical, orconvex spherical. However, other embodiments are possible.

When used in spectacles the lens may comprise front and back surfaceswhich are independently concave or convex, and either one or both ofthese surfaces may be aspheric. Typically, the front surface will beconvex and the back surface will be concave.

Another approach used to correct visual focal problems is surgicalintervention, where the eye is mechanically cut or reshaped by a laser.In particular, excimer laser sculpting methodology is suitable inpracticing the present invention. In this case, the appropriatehyperbolic corneal shape for optimal vision would be determined usingthe method of the present invention, and the shape then produced by thisknown technique. The result would require no additional corrective lens(even for most astigmats or presbyopes) and produce visual acuity betterthan a naturally "perfect" spherical lens.

Although the advantages of the present invention may be obtained in asystem having a single aspheric surface, the present invention alsoincludes the use of multiple aspheric surfaces, either in a single lensor in a combination of lenses.

A lens according to the present invention may be formed from anysuitable high quality optical material, such as optical glass orplastic, but preferably the lens is made of optical quality transparentmolded plastic. Suitable materials also include polymers (includingfluoropolymers), resinous materials, solid or semi-solid gelatinousmaterials, rigid gas permeable materials, and the like. A contact lensconstructed according to the present invention is preferably made of ahydrophilic polymer polmerized from a methacrylate based monomer. A lensaccording to the present invention may be incorporated into spectacles,but the preferred embodiments are contact lenses and intraocular lenses.

Many embodiments and variations of this invention will occur to thoseskilled in the art. The present invention is not limited to theembodiments described and illustrated, but includes every embodimentconsistent with the foregoing description and the attached drawings thatfalls within the scope of the appended claims.

I claim:
 1. A method of constructing a lens for focusing light on theretina of the eye comprising the steps of:a) constructing a FourierTransform function model that generates modulation transfer frequenciesfor the human eye and a preliminary lens, said lens having at least onerotationally symmetric surface defined by the equation: ##EQU4## where Xis the aspheric surface point at position Y, r is the central radius,and κ is commonly used aspheric constant, wherein the value of κ isbetween -0.5 and -1, b) performing an analysis using the model soconstructed to trace light ray paths through the lens-eye system, c)varying the value of the aspheric constant, κ, for the preliminary lensto achieve a lens-eye system with a trace of light ray paths optimizedfor sharpest focus by minimizing retinal spot size of said rays.
 2. Themethod of claim 1 wherein the lens so constructed is a contact lens. 3.The method of claim 1 wherein the modulation transfer frequency iscompared to the diffraction limit to optimize the corrective lens-eyesystem.
 4. The method of claim 1 wherein the eye in the correctivelens-eye system is emmetropic and the optimization process producesvision that exceeds that of the normal eye.
 5. The method of claim 1wherein the corrective lens-eye system is optimized by positioning thefocused image closest to the retina.